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A322746 a(n) = 1/2 * (-1 + Sum_{k=0..n} binomial(2*n,2*k)*(n+1)^(n-k)*n^k). 4

%I

%S 0,1,24,675,25920,1275125,76545000,5425069447,443365544448,

%T 41047124680809,4245890890571000,485307363135371051,

%U 60742714406414040000,8262695239025750162653,1213734518568509516047560,191478489107270936785743375,32288451913272713227175006208

%N a(n) = 1/2 * (-1 + Sum_{k=0..n} binomial(2*n,2*k)*(n+1)^(n-k)*n^k).

%H Seiichi Manyama, <a href="/A322746/b322746.txt">Table of n, a(n) for n = 0..321</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^n.

%F sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^n.

%F a(n) = (A173174(n) - 1)/2.

%F a(n) ~ exp(1/2) * 2^(2*n - 2) * n^n. - _Vaclav Kotesovec_, Dec 25 2018

%e (sqrt(3) + sqrt(2))^2 = 5 + 2*sqrt(6) = sqrt(25) + sqrt(24). So a(2) = 24.

%o (PARI) {a(n) = 1/2*(-1+sum(k=0, n, binomial(2*n,2*k)*(n+1)^(n-k)*n^k))}

%o (PARI) {a(n) = (polchebyshev(n, 1, 2*n+1)-1)/2}

%Y Main diagonal of A322699.

%Y Cf. A322747.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 25 2018

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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)